Multiple scattering by random configurations of circular
cylinders: Second-order corrections for the effective
wavenumber
C. M. Linton
Department of Mathematical Sciences, Loughborough University, Leicestershire LE11 3TU, United Kingdom
P. A. Martin
Department of Mathematical and Computer Sciences, Colorado School of Mines,
Golden, Colorado 80401-1887
~Received 3 September 2004; revised 3 March 2005; accepted 15 March 2005!
A formula for the effective wavenumber in a dilute random array of identical scatterers in two
dimensions is derived, based on Lax’s quasicrystalline approximation. This formula replaces a
widely-used expression due to Twersky, which is shown to be based on an inappropriate choice of
pair-correlation function. © 2005 Acoustical Society of America. @DOI: 10.1121/1.1904270#
PACS numbers: 43.20.Fn, 43.20.Bi, 43.20.Hq, 43.28.En @GCG#
I. INTRODUCTION
Multiple scattering by random arrangements of scatterers is a topic with an extensive literature. See, for example,
the recent book by Tsang et al. ~2001!. The modern era dates
from the work of Foldy ~1945!, Lax ~1951, 1952!, Waterman
and Truell ~1961!, and Twersky ~1962!. Major applications
include wave propagation through suspensions @see, for example McClements et al. ~1990!, Povey ~1997!, Dukhin and
Goetz ~2001!, and Spelt et al. ~2001!# and through elastic
composites @see Mal and Knopoff ~1967!, Kim et al. ~1995!,
and Kanaun ~2000!#. In this paper, we are mainly interested
in two-dimensional problems, motivated by the calculation
of sound propagation through forests @Embleton ~1966!;
Price et al. ~1988!#. An important paper on acoustic scattering by arrays of circular cylinders is that of Bose and Mal
~1973!; see Sec. IV below. For subsequent work, see Varadan
et al. ~1978!, Yang and Mal ~1994!, Bose ~1996!, Kanaun
and Levin ~2003!, and Kim ~2003!. For analogous planestrain elastodynamics, see Varadan et al. ~1986!, Yang and
Mal ~1994!, Bussink et al. ~1995!, and Verbis et al. ~2001!.
A typical problem is the following. The region x,0 is
filled with a homogeneous compressible fluid of density r
and sound-speed c. The region x.0 contains the same fluid
and many scatterers; to fix ideas, we suppose that the scatterers are identical circles ~parallel circular cylinders!. Then,
a time-harmonic plane wave with wavenumber k5 v /c ~v is
the angular frequency! is incident on the scatterers: what is
the reflected wave field? This field may be computed exactly
for any given configuration ~ensemble! of N circles, but the
cost increases as N increases. If the computation can be done,
it may be repeated for other configurations, and then the
average reflected field could be computed ~this is the Monte
Carlo approach!. Instead of doing this, one can try to do
some ensemble averaging in order to calculate the average
~coherent! field. One result of this is a formula for the effective wavenumber K. This can then be used to replace the
‘‘random medium’’ occupying x.0 by a homogeneous effective medium.
Foldy ~1945! began by considering isotropic point scatJ. Acoust. Soc. Am. 117 (6), June 2005
Pages: 3413–3423
terers; this is an appropriate model for small sound-soft scatterers. He obtained the formula
K 2 5k 2 24ign 0 ,
~1!
where n 0 is the number of circles per unit area and g is the
scattering coefficient for an individual scatterer. @In fact,
Foldy considered scattering in three dimensions; the twodimensional formula, Eq. ~1!, can be found as Eq. ~3.20! in
Twersky ~1962! and Eq. ~26! in Aristégui and Angel ~2002!,
for example.# The formula ~1! assumes that the scatterers are
independent and that n 0 is small. We are interested in calculating the correction to Eq. ~1! ~a term proportional to n 20 ),
and this will require saying more about the distribution of the
scatterers; specifically, we shall use pair correlations. Thus,
our goal is a formula of the form
K 2 5k 2 1 d 1 n 0 1 d 2 n 20 ,
~2!
with computable expressions for d 1 and d 2 . Moreover, we
do not only want to restrict our formula to sound-soft scatterers.
There is some controversy over the proper value for d 2 .
In order to state one such formula, we introduce the far-field
pattern f for scattering by one circular cylinder. Thus, we
have u in5exp@ikr cos(u2uin) # for the incident plane wave,
where (r, u ) are plane polar coordinates (x5r cos u, y
5r sin u) and u in is the angle of incidence. The scattered
waves satisfy
u sc; A2/~ p kr ! f ~ u 2 u in! exp~ ikr2ip /4!
as r→`.
~3!
Then, Twersky ~1962! has given the following formula:
K 2 5k 2 24in 0 f ~ 0 ! 1 ~ 2n 0 /k ! 2 sec2 u in$ @ f ~ p 22 u in!# 2
2 @ f ~ 0 !# 2 % .
~4!
This formula involves u in , so that it gives a different effective wavenumber for different incident fields.
The three-dimensional version of Eq. ~4! is older. For a
random collection of identical spheres, it is
0001-4966/2005/117(6)/3413/11/$22.50
© 2005 Acoustical Society of America
3413
K 2 5k 2 24 p i~ n̂ 0 /k ! f ~ 0 ! 1 d 2 n̂ 20
with @see Twersky ~1962!#
d 2 5 ~ 4 p 2 /k 4 ! sec2 u in$ @ f ~ p 22 u in!# 2 2 @ f ~ 0 !# 2 % ,
~5!
where the far-field pattern is now defined by u sc
;(ikr) 21 eikr f ( q ), r and q are spherical polar coordinates,
and n̂ 0 is the number of spheres per unit volume. The same
formula but with u in50 ~normal incidence! was given by
Waterman and Truell ~1961!. However, it was shown by
Lloyd and Berry ~1967! that Eq. ~5! is incorrect; they obtained
d 25
H
4p2
k
1
4
E
p
0
2 @ f ~ p !# 2 1 @ f ~ 0 !# 2
d
1
@ f ~ q !# 2 dq
sin~ q /2! dq
J
~6!
~with no dependence on u in). Lloyd and Berry ~1967! used
methods ~and language! coming from nuclear physics. Thus,
in their approach, which they ‘‘call the ‘resummation
method,’ a point source of waves is considered to be situated
in an infinite medium. The scattering series is then written
out completely, giving what Lax has called the ‘expanded’
representation. In this expanded representation the ensemble
average may be taken exactly @but then# the coherent wave
does not exist; the series must be resummed in order to obtain any result at all.’’ One purpose of the present paper is to
demonstrate that a proper analysis of the semi-infinite twodimensional model problem ~with arbitrary angle of incidence! leads to a formula that is reminiscent of the ~threedimensional! Lloyd–Berry formula; specifically, instead of
Eq. ~4!, we obtain
K 2 5k 2 24in 0 f ~ 0 ! 1
8n 20
pk2
E
p
0
d
cot~ u /2! @ f ~ u !# 2 du .
du
~7!
Our analysis does not involve ‘‘resumming’’ series or divergent integrals. It builds on a conventional approach, in the
spirit of the papers by Fikioris and Waterman ~1964! and by
Bose and Mal ~1973!.
The paper is organized, as follows. Some elementary
probability theory is recalled in Sec. II. In particular, the
pair-correlation function is introduced; this leads to the notion of ‘‘hole correction’’—individual cylinders must not be
allowed to overlap during the averaging process. In Sec. III,
we derive the integral equations of Foldy ~isotropic scatterers, no hole correction! and of Lax ~isotropic scatterers, hole
correction included!. Foldy’s integral equation can be solved
exactly whereas we have been unable to solve Lax’s integral
equation. Nevertheless, we have developed a rigorous
method for extracting an expression for K from these integral
equations without actually solving the integral equations
themselves. Then, we use the same method in Sec. IV but
without the restriction to isotropic scatterers. We start by
following Bose and Mal ~1973!, and use an exact ~deterministic! theory for scattering by N circles followed by ensemble
averaging. We give a clear derivation of a certain homogeneous infinite linear system of algebraic equations, obtained
3414
J. Acoust. Soc. Am., Vol. 117, No. 6, June 2005
previously by Bose and Mal ~1973! for the case of normal
incidence; the system does not depend on u in and the existence of a nontrivial solution determines K. We solve the
system for small n 0 , and obtain Eq. ~7!. We also show that
Eq. ~4! is obtained if the hole correction is not done correctly. Concluding remarks are given in Sec. V.
II. SOME PROBABILITY THEORY
In this section, we give a very brief summary of the
probability theory needed. For more information, see Foldy
~1945!, Lax ~1951!, Aristégui and Angel ~2002! or Chap. 14
of Ishimaru ~1978!.
Suppose we have N scatterers located at the points r1 ,
r2 ,...,rN ; denote the configuration of points by L N
5 $ r1 ,r2 ,...,rN % . Then, the ensemble ~or configurational! average of any quantity F(ru L N ) is defined by
^ F ~ r! & 5
E E
¯
p ~ r1 ,r2 ,...,rN ! F ~ ru L N ! dV 1 ¯dV N ,
~8!
where the integration is over N copies of the volume B N
containing N scatterers. Here, p(r1 ,...,rN )dV 1 dV 2 ¯dV N is
the probability of finding the scatterers in a configuration in
which the first scatterer is in the volume element dV 1 about
r1 , the second scatterer is in the volume element dV 2 about
r2 , and so on, up to rN . The joint probability distribution
p(r1 ,...,rN ) is normalized so that ^1&51. Similarly, the average of F(ru L N ) over all configurations for which the first
scatterer is fixed at r1 is given by
^ F ~ r! & 1 5
E E
¯
p ~ r2 ,...,rN u r1 ! F ~ ru L N ! dV 2 ¯dV N ,
~9!
where p(r1 ,r2 ,...,rN )5p(r1 ) p(r2 ,...,rN u r1 ) defines the
conditional probability p(r2 ,...,rN u r1 ). If two scatterers are
fixed, say the first and the second, we can define
^ F ~ r! & 125
E E
¯
p ~ r3 ,...,rN u r1 ,r2 ! F ~ ru L N ! dV 3 ¯dV N ,
~10!
where p(r2 ,...,rN u r1 )5p(r2 u r1 )p(r3 ,...,rN u r1 ,r2 ).
Now, as each of the N scatterers is equally likely to
occupy dV 1 , the density of scatterers at r1 is N p(r1 )5n 0 ,
the ~constant! number of scatterers per unit volume. Thus
p ~ r! 5n 0 /N5 u B N u 21 ,
~11!
where u B N u is the volume of B N . Also, as p(r1 ,r2 )
5p(r1 )p(r2 u r1 ), we obtain
EE
p ~ r2 u r1 ! dV 1 dV 2 5
N
5 u B Nu .
n0
~12!
We have to specify p(r2 u r1 ), consistent with Eq. ~12!. Also,
we want to ensure that scatterers do not overlap. For circular
cylinders of radius a, a simple choice is p(r2 u r1 )
5p 0 H(R 122b) with b>2a, where H(x) is the Heaviside
unit function, R 125 u r1 2r2 u and p 0 is a constant determined
by Eq. ~12!. Thus,
p 0 5 $ u B N u 2 p b 2 % 21 .n 0 /N,
~13!
C. Linton and P. Martin: Scattering by random arrangements of cylinders
assuming that b 2 n 0 /N!1. @The equality in Eq. ~13! assumes
that the ‘‘hole’’ at r1 of radius b does not cut the boundary of
B N . Evidently, taking this possibility into account would not
change the approximation p 0 .n 0 /N.] Hence, the simplest
sensible choice for the pair-correlation function is
p ~ r2 u r1 ! 5
H
0,
R 12,b,
n 0 /N,
~14!
R 12.b.
This simple choice will be used for most of our analysis.
More generally, we could use
p ~ r2 u r1 ! 5
H
0,
R 12,b,
~ n 0 /N !@ 11 x ~ R 12 ;n 0 !# ,
R 12.b,
~15!
where the function x is to be chosen, subject to some constraints. The effect of using Eq. ~15! instead of Eq. ~14! is
calculated in Sec. IV D. One could also consider functions x
that depend on r1 2r2 ~instead of just u r1 2r2 u ); such possibilities are discussed in Twersky ~1978! and Siqueira et al.
~1995!.
III. FOLDY–LAX THEORY: ISOTROPIC SCATTERERS
Foldy’s theory begins with a simplified deterministic
model for scattering by N identical scatterers, each of which
is supposed to scatter isotropically. Thus, the total field is
assumed to be given by the incident field plus a point source
at each scattering center, r j :
N
u ~ ru L N ! 5u in~ r! 1g
(
j51
u ex~ r j ;r j u L N ! H 0 ~ k u r2r j u ! .
~16!
Here, H n (w)[H (1)
n (w) is a Hankel function, g is the ~assumed known! scattering coefficient, and the exciting field
u ex is given by
N
u ex~ r;rn u L N ! 5u in~ r! 1g
(
j51
jÞn
u ex~ r j ;r j u L N ! H 0 ~ k u r2r j u ! .
~17!
The second term in Eq. ~17! is the field near the cylinder at
rn due to scattering by all the other cylinders. The N numbers
u ex(r j ;r j u L N ) ( j51,2,...,N) required in Eq. ~16! are to be
determined by solving the linear system obtained by evaluating Eq. ~17! at r5rn ; direct numerical solutions of this
system have been given by Fikioris ~1966! and by Groenenboom and Snieder ~1995!.
Let us try to compute the ensemble average of u, using
Eqs. ~16! and ~8!. The result is
^ u ~ r! & 5u in~ r! 1gn 0
E
BN
^ u ex~ r! & 1 5u in~ r! 1g ~ N21 !
where we have used Eqs. ~9! and ~11!, and the indistinguishability of the scatterers. For ^ u ex(r1 ) & 1 @which is given explicitly by Eq. ~9! in which u ex(r1 ;r1 u L N ) is substituted for
F(ru L N )], we obtain
J. Acoust. Soc. Am., Vol. 117, No. 6, June 2005
BN
p ~ r2 u r1 !
3^ u ex~ r2 ! & 12H 0 ~ k u r2r2 u ! dV 2 ,
~19!
where we have used Eqs. ~10! and ~17!. Equations ~18! and
~19! are the first two in a hierarchy, involving more and more
complicated information on the statistics of the scatterer distribution. In practice, the hierarchy is broken using an additional assumption. At the lowest level, we have Foldy’s assumption,
^ u ex~ r! & 1 . ^ u ~ r! & ,
~20!
at least in the neighborhood of r1 . When this is used in Eq.
~18!, we obtain
^ u ~ r! & 5u in~ r! 1gn 0
E
BN
^ u ~ r1 ! & H 0 ~ k u r2r1 u ! dV 1 ,
~21!
rPB N .
We call this Foldy’s integral equation for ^ u & . The integral
on the right-hand side is an acoustic volume potential.
Hence, an application of (¹ 2 1k 2 ) to Eq. ~21! eliminates the
incident field and shows that (¹ 2 1K 2 ) ^ u & 50 in B N , where
K 2 is given by Foldy’s formula, Eq. ~1!.
At the next level, we have the Lax ~1952! quasicrystalline assumption ~QCA!,
^ u ex~ r! & 12. ^ u ex~ r! & 2 .
~22!
When this is used in Eq. ~19! evaluated at r5r1 , we obtain
v~ r! 5u in~ r! 1g ~ N21 !
E
BN
p ~ r1 u r!v~ r1 ! H 0 ~ k u r2r1 u ! dV 1 ,
~23!
rPB N ,
where v (r)5 ^ u ex(r) & 1 . We call this Lax’s integral equation.
In what follows, we let N→` so that B N →B ` , a semiinfinite region, x.0.
A. Foldy’s integral equation: Exact treatment
Consider a plane wave at oblique incidence, so that
u in5ei~ a x1 b y !
with a 5k cos u in and b 5k sin u in .
~24!
For a semi-infinite domain B ` , Foldy’s integral equation,
Eq. ~21!, becomes
^ u ~ x,y ! & 5ei~ a x1 b y ! 1gn 0
EE
`
0
3H 0 ~ k r 1 ! dY dx 1 ,
^ u ex~ r1 ! & 1 H 0 ~ k u r2r1 u ! dV 1 ,
~18!
E
`
2`
^ u ~ x 1 ,y1Y ! &
x.0,
2`,y,`,
where r 1 5 A(x2x 1 ) 2 1Y 2 . This equation can be solved exactly. Thus, writing
^ u ~ x,y ! & 5U ~ x ! eim y ,
x.0,
2`,y,`,
~25!
we obtain
C. Linton and P. Martin: Scattering by random arrangements of cylinders
3415
U ~ x ! 5eia x ei~ b 2 m ! y 1gn 0
EE
`
0
U 0 eilx 2eia x
`
2`
3H 0 ~ k r 1 ! eim Y dY dx 1 ,
U~ x1!
x.0,
2`,y,`.
5
~26!
Hence, for a solution in the form Eq. ~25!, we must have m
5 b 5k sin uin .
Now,
E
`
2`
ib Y
H 0~ k r 1 ! e
2
dY 5 eia u x2x 1 u ,
a
~27!
where a 5 Ak 2 2 b 2 5k cos uin . Thus, we see that U solves
U ~ x ! 5eia x 1
2gn 0
a
E
`
0
U ~ x 1 ! eia u x2x 1 u dx 1 ,
x.0,
2`,y,`,
~29!
`
,
U ~ t ! e2ia t dt1
0
2gn 0
a
U ~ t ! eia u x2t u dt5Aeilx 1Beia x
for x.,,
where A524ign 0 U 0 /(l 2 2 a 2 ) and
B5
2gn 0
a
E
,
U ~ t ! e2ia t dt1
0
2ign 0 U 0 i~ l2 a ! ,
e
.
a ~ l2 a !
Then, setting U 0 5A gives Eq. ~30! again, without knowing
the solution U everywhere. This basic method will be used
again below.
D
Using the approximation p(r1 u r)5(n 0 /N)H(R 1 2b) in
Lax’s integral equation, Eq. ~23! gives
N21
N
v~ r! 5u in~ r! 1gn 0
eia x
2gn 0 U 0 2 a eilx
,
2
U 0 eilx 2eia x 5
ia
l 2 2 a 2 l2 a
where we have assumed that Im l.0. If we compare the
coefficients of eilx , we see that U 0 cancels, leaving
l 2 2 a 2 524ign 0 ,
~30!
ia x
which determines l. Then, the coefficients of e give U 0
52 a /(l1 a ). A similar method can be used to find ^ u &
when B ` is a slab of finite thickness, 0,x,h, say; see
Aristégui and Angel ~2002!.
From Eq. ~29!, it is natural to write
l5K cos w
E
,
C. Lax’s integral equation
and Eq. ~28! gives
S
3
E
x.0. ~28!
Now, put U(x)5U 0 eilx , so that Eq. ~25! gives
^ u ~ x,y ! & 5U 0 ei~ lx1 b y ! ,
2gn 0 ia x
e
a
and b 5K sin w 5k sin u in .
~31!
These define the effective wave number K; the last equality
is recognized as Snell’s law, even though K and w are complex, with Im K.0. Hence, we see that
l 2 2 a 2 5K 2 2k 2 ,
~32!
E
b
B N ~ r!
v~ r1 ! H 0 ~ kR 1 ! dr1 ,
~33!
rPB N ,
B Nb (r)5 $ r1 PB N :R 1 5 u r2r1 u .b % ,
where
which is B N with
a ~possibly incomplete! disk excluded.
Let N→` and take an incident plane wave, Eq. ~24!,
giving
v~ x,y ! 5ei~ a x1 b y ! 1gn 0
E
x 1 .0,r 1 .b
v~ x 1 ,y1Y !
x.0,
2`,y,`.
3H 0 ~ k r 1 ! dY dx 1 ,
As in Sec. III A, we write
v~ x,y ! 5V ~ x ! eib y ,
x.0,
~34!
2`,y,`,
giving
V ~ x ! 5eia x 1gn 0
and so Eq. ~30! reduces to Foldy’s formula, Eq. ~1!.
E
x 1 .0,r 1 .b
3eib Y dY dx 1 ,
V ~ x 1 ! H 0~ k r 1 !
~35!
x.0.
Then, using Eq. ~27!, we see that V solves
B. Foldy’s integral equation: Approximate treatment
We have seen that Foldy’s integral equation can be
solved exactly, and that the solution process has two parts:
first find l ~and hence the effective wavenumber! and then
find U 0 . In fact, l can be found without finding the complete
solution; the reason for pursuing this is that we cannot usually find exact solutions. Thus, consider Eq. ~28!, and suppose that
U ~ x ! 5U 0 eilx
for x.,,
where U 0 , l, and , are unknown. To proceed, we need say
nothing about the solution U in the ‘‘boundary layer’’ 0,x
,,. Now, evaluate the integral equation for x.,; we find
that
3416
J. Acoust. Soc. Am., Vol. 117, No. 6, June 2005
V ~ x ! 5eia x 1gn 0
E
`
0
V ~ x 1 ! L ~ x2x 1 ! dx 1 ,
x.0,
~36!
where the kernel, L(x2x 1 ), is given by
L~ X !5
2 ia u X u
e
22
a
E
c~ X !
0
H 0 ~ k AX 2 1Y 2 ! eib Y dY
~37!
with c(X)5 Ab 2 2X 2 H(b2 u X u ); here, we have written the
integral over Y in Eq. ~35! as an integral over all Y minus an
integral through the disk, if necessary.
We have been unable to solve Eq. ~36! exactly ~even
though it is an integral equation of Wiener–Hopf-type!.
However, the approximate method described in Sec. III B
can be used. Thus, let us suppose that
C. Linton and P. Martin: Scattering by random arrangements of cylinders
V ~ x ! 5V 0 eilx
~38!
for x.,,
where V 0 , l, and , are unknown. Then, consider Eq. ~36! for
x.,1b, so that the interval u x2x 1 u ,b is entirely within
the range x 1 .,. Making use of Eq. ~37!, Eq. ~36! gives
V 0 eilx 2eia x
gn 0
5
2 ia x
e
a
22
E
E
,
V ~ t ! e2ia t dt1
0
x1b
V~ t !
x2b
E
c ~ x2t !
0
2
a
E
`
V ~ t ! eia u x2t u dt
,
H 0 ~ k A~ x2t ! 2 1Y 2 ! eib Y dY dt
FIG. 1. A view of two typical cylinders.
~39!
for x.,1b. Equation ~38! can be used in the second and
third integrals. The second integral is elementary, and has the
value
where d 0 (x)5J 0 (x)H 0 (x)1J 1 (x)H 1 (x). When this approximation for N0 (Kb) is used in Eq. ~40!, we obtain
The third integral becomes
22V 0
E
E
E E
E
Ab 2 2 j 2
il ~ x1 j !
e
0
2b
52V 0 eilx
2p
0
0
522 p V 0 eilx
5V 0 eilx
H
b
2
ib Y
K 2 5k 2 24ign 0 2 p b 2 g d 1 d 0 ~ kb ! n 20 .
dY dj
Comparison of this formula with Eq. ~41! gives d 1 524ig
~as expected! and d 2 54 p i(gb) 2 d 0 (kb), so that we obtain
the approximation
eiKr cos~ u 2 w ! H 0 ~ kr ! rdrdu
K 2 5k 2 24ign 0 14 p i~ gbn 0 ! 2 d 0 ~ kb ! .
b
0
K 2k
2
2
2 p N0 ~ Kb !
K 2 2k 2
J
,
IV. FINITE-SIZE EFFECTS
where
N0 (Kb)5KbH 0 (kb)J 1 (Kb)2kbH 1 (kb)J 0 (Kb).
Using these results in Eq. ~39!, noting Eq. ~32!, we obtain
V 0 eilx 2eia x 5Aeilx 1Beia x
for x.,1b,
where
A5
B5
2 p gn 0 V 0
k 2 2K 2
2gn 0
a
E
N0 ~ Kb ! ,
,
V ~ t ! e2ia t dt1
0
2ign 0 V 0 i~ l2 a ! ,
e
.
a ~ l2 a !
For a solution, we must have A5V 0 , and so
K 2 5k 2 22 p gn 0 N0 ~ Kb ! ,
~40!
which is a nonlinear equation for K. Notice that this equation
does not depend on the angle of incidence, u in .
We have N0 (Kb)→2i/ p as b→0 so that, in this limit,
we recover Foldy’s formula for the effective wavenumber,
Eq. ~1!.
Let us solve Eq. ~40! for small n 0 . ~Alternatively, we
could use the dimensionless area fraction p a 2 n 0 .) Begin by
writing
K 2 5k 2 1 d 1 n 0 1 d 2 n 20 1¯,
J. Acoust. Soc. Am., Vol. 117, No. 6, June 2005
~42!
Note that the second-order term in Eq. ~42! vanishes in the
limit kb→0.
J 0 ~ Kr ! H 0 ~ kr ! rdr
4i
2
H 0 ~ k Aj 1Y ! e
2
N0 ~ Kb ! 5N0 ~ kb ! 1 ~ Kb2kb ! N 80 ~ kb ! 1¯
52i/ p 1 21 b 2 d 1 d 0 ~ kb ! n 0 1O ~ n 20 ! ,
4iV 0 ilx
2iV 0
ei~ l2 a ! , eia x 2 2
e .
a ~ l2 a !
l 2a2
b
where d 1 and d 2 are to be found; for d 1 , we expect to obtain
the result given by Eq. ~1!. It follows from Eq. ~41! that K
5k1 12 d 1 n 0 /k1O(n 20 ) and then
~41!
The theory described above relies on the assumption of
isotropy. Here, we use a more complete theory. We start with
an exact theory ~due to Záviška! for acoustic scattering by N
identical circular cylinders of radius a; for details and references, see p. 173 of Linton and McIver ~2001!. The cylinders
can be soft, hard or penetrable. Then ~in Sec. IV B!, we form
averaged equations, and we invoke the QCA. This leads to
an infinite homogeneous system of linear algebraic equations
from which the effective wave number, K, is to be determined; the equations are independent of the angle of incidence. An approximate solution for K is found in Sec. IV C,
correct to O(n 20 ). In Sec. IV D, it is shown that this approximation does not depend on the choice of the function
x (r;n 0 ), appearing in the pair-correlation function, Eq. ~15!.
In Sec. IV E, it is shown how Twersky’s formula for K can be
derived, using an unreasonable choice for the pair-correlation
function.
A. A finite array of identical circular cylinders: Exact
theory
We use polar coordinates (r, u ) centered at the origin
and (r j , u j ), centered at r j 5(x j ,y j ), the center of the jth
cylinder. The various parameters relating to the relative positions of the cylinders are shown in Fig. 1.
Exterior to the cylinders the pressure field is u, where
C. Linton and P. Martin: Scattering by random arrangements of cylinders
3417
¹ 2 u1k 2 u50.
~43!
In the interior of cylinder j, the field is u j , where
¹ 2 u j 1 k 2 u j 50.
~44!
A plane wave, given by Eq. ~24!, is incident on the cylinders.
A phase factor for each cylinder, I j , is defined by
i~ a x j 1 b y j !
~45!
I j 5e
and then we can write
`
ikr j cos~ u j 2 u in!
u in5I j e
5I j
(
n52`
ein ~ p /22 u j 1 u in! J n ~ kr j ! .
~46!
We seek a solution to Eqs. ~43! and ~44! in the form
`
N
u5u in1
( (
j51 n52`
A nj Z n H n ~ kr j ! ein u j ,
~47!
`
u j5
(
n52`
B nj J n ~ k r j ! ein u j ,
~48!
for some set of unknown complex coefficients A nj and B nj .
The factor
Z n5
qJ 8n ~ ka ! J n ~ k a ! 2J n ~ ka ! J 8n ~ k a !
qH 8n ~ ka ! J n ~ k a ! 2H n ~ ka ! J 8n ~ k a !
5Z 2n
~49!
has been introduced for later convenience. Here k 5 v /c̃ and
q5 r̃ c̃/( r c), where r̃ and c̃ are the density and sound speed,
respectively, inside the cylinders. Note that we recover the
sound-soft results in the limit q→0, whereas the limit q
→` gives the sound-hard results. The boundary conditions
on the cylinders are
u5u s ,
1 ]u 1 ]us
5
r ] r s r̃ ] r s
on r s 5a,
N
1
~ I s J n ~ kr s ! ein ~ p /22 u s 1 u in! 1A sn Z n H n ~ kr s ! ein u s !
`
( (
j51 n52`
jÞs
`
A nj Z n
(
m52`
J m ~ kr s ! H n2m ~ kR js !
3eim u s ei~ n2m ! a js .
~52!
`
f ~ u ! 52
(
Z n ein u .
n52`
~53!
B. Arrays of circular cylinders: Averaged equations
The above analysis applies to a specific configuration of
scatterers. Now we follow Bose and Mal ~1973! and take
ensemble averages. Specifically, setting s51 in Eq. ~52! and
then taking the conditional average, using Eq. ~14!, we get
`
^ A m1 & 1 1n 0
N21
Z
N n52` n
(
E
B N :R 12.b
H n2m ~ kR 21!
3ei~ n2m ! a 21^ A 2n & 12dV 2
52I 1 eim ~ p /22 u in! ,
~54!
mPZ.
Now we let N→` so that B N becomes the half-space
x.0, and invoke Lax’s QCA, Eq. ~22!. This implies that
^ A m2 & 125 ^ A m2 & 2 .
~55!
We seek a solution to Eq. ~54! in the form
^ A ms & s 5im eib y s F m ~ x s !
~56!
`
J. Acoust. Soc. Am., Vol. 117, No. 6, June 2005
(
n52`
Z n ~ 2i! n2m
E
x 2 .0,R 12.b
c n2m ~ x 21 ,y 21!
3eib y 21F n ~ x 2 ! dx 2 dy 2
52e2im u ineia x 1 ,
~57!
mPZ,
where we have written x 215x 2 2x 1 and y 215y 2 2y 1 , used
a 215 a 122 p , and defined c n (X,Y )5H n (kR)einQ with X
5R cos Q and Y 5R sin Q.
Proceeding as before, suppose that for sufficiently large
x ~say x.,) we can write
F m ~ x ! 5F m e2im w eilx ,
~51!
The geometrical restriction implies that this expression is
only valid if the point (r s , u s ) is closer to the center of cylinder s than the centers of any of the other cylinders. This is
certainly true on the surface of cylinder s and so Eq. ~51! can
be used to apply the body boundary conditions which leads,
after using the orthogonality of the functions exp(im u s ), m
PZ, and eliminating the coefficients B nj , to the system of
equations
3418
s51,2,...,N,
mPZ.
Note that the quantities q, k, and a only enter the equations
through the terms Z n .
1
For a single cylinder the solution is immediate: A m
m
52i I 1 exp(2im u in) and then the far-field pattern, defined
by Eq. ~3!, is given by
F m ~ x 1 ! 1n 0
`
(
A nj Z n ei~ n2m ! a js H n2m ~ kR js !
52I s eim ~ p /22 u in! ,
s51,...,N. ~50!
u~ rs ,us!
n52`
( (
j51 n52`
jÞs
so that
Using Graf’s addition theorem for Bessel functions, it
can be shown that provided r s ,R js for all j, we can write the
field exterior to cylinder s as
5
`
N
s
Am
1
~58!
where l and w are defined by Eq. ~31!. We assume that Im
l.0 so that F m →0 as x→`. Then if x 1 .,1b, Eq. ~57!
becomes
`
F m e2im w eilx 1 1n 0
3
HE
(
n52`
Z n ~ 2i! n2m
,
0
F n ~ x 2 ! L n2m ~ x 21! dx 2 1F n e2in w eilx 1 M n2m
52e2im u ineia x 1 ,
mPZ,
J
~59!
C. Linton and P. Martin: Scattering by random arrangements of cylinders
The contribution from the circle R 125b is
where
L n~ X ! 5
M n5
E
`
2`
E
~60!
c n ~ x 21 ,y 21! C ~ x 21 ,y 21! dx 2 dy 2 ,
x 2 .,,R 12.b
E F c ]]
E
2p
c n ~ X,Y ! eib Y dY ,
C ~ X,Y ! 5ei~ lX1 b Y ! 5eiKR cos~ Q2 w ! ,
~61!
2
E
S
2 ~ ia 2 b !
ak
n
D
3
2i in u 2ia X
e ine
.
a
L n ~ x 2 2x 1 ! 5 ~ 2/a ! i e
1
k 2K
2
E Fc
]B
E F c ]]
x 2 5,
e
2
m
G
]C
]cm
2C
ds 2 ,
]n
]n
G
F
E
`
F
x.,1b,
~69!
mPZ,
where
Am 5F m 1
2n 0 p
`
(
k 2 2K 2 n52`
F n Z n Nn2m ~ Kb ! ,
2n 0
Z ein u in
B5
a n52` n
2`
E
`
G
]cn
] R 12
dy 2
eib y 21
2`
G
dy 2
x 2 5,
2
5 ei~ a 2l !~ x 1 2, ! in21 ein u in~ l1 a ! ,
a
using
2H n8 (x)5H n21 (x)2H n11 (x),
5H n21 (x)1H n11 (x), and Eq. ~66! thrice.
J. Acoust. Soc. Am., Vol. 117, No. 6, June 2005
HE
,
0
F n ~ t ! e2ia t dt1
J
iF n e2in w i~ l2 a ! ,
e
,
l2 a
and
Nn ~ Kb ! 5kbH n8 ~ kb ! J n ~ Kb ! 2KbH n ~ kb ! J n8 ~ Kb ! . ~70!
x 2 5,
k
3 2il c n 1 ~ c n21 2 c n11 !
2
3
In particular, note that N0 appeared in Sec. III C during our
analysis of Lax’s integral equation.
From Eq. ~69!, we immediately obtain B521 and Am
50 for all m; the second of these, namely
F m1
eib y 21 2il c n 1cos a 12
sin a 12 ] c n
R 12 ] a 12
5eil ~ ,2x 1 !
G
iK
H ~ kb !~ eiu 1e2iu ! 2kH n8 ~ kb ! du
2 n
52e2im u ineia x ,
~66!
.
C
]cn
dy 2
2C
x2
]x2
5eil ~ ,2x 1 !
iq J q ~ Kb ! e2iq u
(
where ] B consists of two parts, the line x 2 5, and the circle
R 125b. Now, on x 2 5,, ] / ] n52 ] / ] x 2 and so we have
n
(
q52`
0
Am e2im w eilx 1Be2im u ineia x
~65!
The double integral M n can be evaluated using Green’s
theorem as follows. We have c n ¹ 2 C2C¹ 2 c n 5(k 2
2K 2 ) c n C. It follows that
2
`
ein u
`
n in u in ia ~ x 1 2x 2 !
2
F
E
2p
Thus, the system ~59! can be written as
~64!
This formula also holds for n,0. Hence, for x 1 .x 2 ,
M m5
eiKb cos~ Q2 w ! einQ
522 p bin ein w @ KH n ~ kb ! J 8n ~ Kb ! 2kH n8 ~ kb ! J n ~ Kb !# .
n
e2ia X 5
bdQ
R5b
~68!
Then, as the ] / ] X can be taken outside the integral and the
] / ] Y can be removed using an integration by parts, we have
8 2 b L n21 , which expresses L n in terms of L n21
kL n 52L n21
8 . It follows from Eq. ~63! that
and L n21
L n5
G
0
52bein w
~63!
1
]
]
c ~ X,Y ! dY .
2 eib Y
1i
L n~ X ! 5
k
]X
] Y n21
2`
n
]cn
]R
3 @ iKH n ~ kb ! cos~ Q2 w ! 2kH n8 ~ kb !# dQ
For L n with n.0, we use the fact that
`
R
~ eiKR cos~ Q2 w ! ! 2eiKR cos~ Q2 w !
2p
~62!
and L 08 52ia L 0 .
L 0 ~ X ! 5 ~ 2/a ! e
0
52b
and we have used Eq. ~31!. Next, we shall evaluate L n and
M n ; note that we have x 2 ,,,x 1 in Eq. ~59! so that x 21
,0.
Consider the integral L n (X) for X,0. From Eq. ~27!,
we have
2ia X
n
~67!
(2n/x)H n (x)
2n 0 p
`
(
k 2 2K 2 n52`
F n Z n Nn2m ~ Kb ! 50,
mPZ,
~71!
is of most interest to us. It is an infinite homogeneous system
of linear algebraic equations for F m , mPZ. The existence of
a nontrivial solution to Eq. ~71! determines K. Notice that
Eq. ~71! does not depend on u in , so that the effective wavenumber cannot depend on u in .
Equation ~71! is the same as Eq. ~33! in Bose and Mal
~1973! @with the choice Eq. ~14!#; these authors began by
considering normal incidence, u in50. However, the derivation of Eq. ~71! given here has some advantages over that
given by Bose and Mal ~1973!. First, we do not invoke ‘‘the
so-called ‘extinction theorem’’’ of Lax; this is described in
Sec. VI of Lax ~1952!. Roughly speaking, this ‘‘theorem’’
asserts that one may simply delete the incident field when
calculating the effective wavenumber, in the limit N→`.
Along with this come some divergent integrals; for example,
the integrals in the unnumbered equation between Eqs. ~32!
C. Linton and P. Martin: Scattering by random arrangements of cylinders
3419
and ~33! of Bose and Mal ~1973! are divergent, because eiKx
is exponentially large as x→2`. In fact, we can say that our
analysis proves Lax’s theorem in our particular case.
Second, when dealing with a half-space containing scatterers, we know from the work of Lloyd and Berry ~1967!
that the boundary of the half-space can cause difficulties.
Here, we give a proper treatment of this boundary. In particular, we do not assume that all fields are proportional to
eilx everywhere inside the half-space, x.0, but only in x
.,, away from the boundary: the width of the boundary
layer, ,, is not specified, and need not be specified if one
only wants to calculate K.
A more recent analysis was given by Siqueira and Sarabandi ~1996!. They allow noncircular and nonidentical cylinders ~using a T-matrix formulation! but they do assume
that the effective field is proportional to eilx for all x.0.
2
Nn ~ Kb ! 52i/ p 1 21 b 2 d 1 d n ~ kb ! n 0 1O ~ n 20 ! ,
~72!
`
(
(
iF d 2
,
4 f ~0!
`
Q52
1
iF d 2
Z nq n2
f ~ 0 ! n52`
4 f ~0!
(
S
`
52
2
(
and so
Nn ~ Kb !
k 2 2K 2
52
b 2 d n ~ kb ! 2id 2
2i
1
2
1O ~ n 0 ! . ~74!
p d 1n 0
2
p d 21
If Eq. ~74! is substituted in Eq. ~71! and O(n 20 ) terms
neglected we get
F m5
`
4i
`
(
d 1 n52`
S
Z n F n 1n 0
2
(
Z nF n
4id 2
D
n52`
3 p b d n2m ~ kb ! 2
d 21
`
`
d 2 54 p ib
2
( (
n52` s52`
`
( (
n52` s52`
Z n Z s d s2n ~ kb ! 1¯.
~81!
For isotropic point scatterers, we have u Z 0 u @ u Z n u for all
nÞ0 and g52Z 0 , so that Eq. ~81! reduces to Eq. ~42! in
this limit.
So far we have not made any assumptions about the size
of ka or kb ~though clearly kb>2ka). Now we will assume
that kb is small. In the limit x→0, we have x 2 d n (x)
;2iu n u / p . Hence as kb→0,
d 2 ;2
8
`
`
( s52`
( u s2n u Z n Z s .
k 2 n52`
~82!
Now
mPZ.
~75!
@ f ~ u !# 2 5
`
( (
n52` s52`
`
5
Z n Z s ei~ n1s ! u
`
( (
n52` s52`
Z n Z s cos~ n2s ! u
~83!
`
4i
(
d 1 n52`
Z nF n ,
mPZ,
~76!
`
( Z s 524if ~ 0 ! ,
s52`
d 1 54i
d
@ f ~ u !# 2 52
du
n52`
`
( s52`
( ~ n2s ! Z n Z s sin~ n2s ! u .
~84!
Also
~77!
where f is the far-field pattern, given by Eq. ~53!.
Returning to Eq. ~75!, we now put F m 5F1n 0 q m , and
then the O(n 0 ) terms give
J. Acoust. Soc. Am., Vol. 117, No. 6, June 2005
since Z n 5Z 2n . Thus
`
which implies that all the F m are equal. If we write F m
5F, Eq. ~76! becomes
3420
~80!
K 2 5k 2 24in 0 f ~ 0 !
At leading order this gives
F m5
Z n Z s d s2n ~ kb !
and so we obtain the approximation
`
,
D
~79!
`
~73!
(
iF d 2
.
4 f ~0!
14 p ib 2 n 20
d n ~ x ! 5J 8n ~ x ! H n8 ~ x ! 1 @ 12 ~ n/x ! # J n ~ x ! H n ~ x !
`
1
Z Q1F p b 2
Z s d s2n ~ kb !
f ~ 0 ! n52` n
s52`
Hence
where
2
~78!
mPZ.
`
Z n d n2m must be indepenIt follows that q m 2 p b 2 F ( n52`
dent of m, call it Q:
C. Approximate determination of K for small n 0
The only approximation made in the derivation of Eq.
~71! is the QCA, which is expected to be valid for small
values of the scatterer concentration (n 0 a 2 !1). We now assume ~as in Sec. III C! that n 0 /k 2 is also small and write
K 2 5k 2 1 d 1 n 0 1 d 2 n 20 1¯. We then have
`
1
q m 52
Z q 1 p b 2F
Z n d n2m ~ kb !
f ~ 0 ! n52` n n
n52`
E
p
0
cot 21 u sin m u du 5 p sgn~ m ! ,
~85!
see Eq. 3.612~7! in Gradshteyn and Ryzhik ~2000!. Thus,
setting kb50 gives
C. Linton and P. Martin: Scattering by random arrangements of cylinders
K 2 5k 2 24in 0 f ~ 0 ! 1
8n 20
pk2
E
p
cot~ u /2 !
0
d
@ f ~ u !# 2 du .
du
~86!
The integral appearing here is convergent because f 8 (0)
50.
N p ~ r2 u r1 ! 5
b8
b
F m 12n 0 p
(
n52`
F nZ n
( Z n~ 2i! n2m
n52`
H
k 2 2K 2
SE E D
`
x 1 2a
1
0
x 1 1a
mPZ,
~89!
where L n (X) is defined by Eq. ~60!.
Suppose that for x., we can write @cf. Eq. ~58!#
F m ~ x ! 5F m e2im u ineilx ,
~90!
where Im l.0. Then if x 1 .,1a, Eq. ~89! becomes
`
F m eilx 1 1n 0
( Z n~ 2i! n2m eim u
n52`
in
E
,
0
F n ~ x 2 ! L n2m ~ x 21!
`
(
ilx 1
3dx 2 1n 0 e
n52`
SE E D
`
1
,
x 1 1a
F n Z n e2i~ n2m !~ p /21 u in!
L n2m ~ x 21! eilx 21dx 2 52eia x 1 ,
~91!
a L n~ x ! 5
and M n is defined by Eq. ~61!. Hence, we obtain a modified
form of Eq. ~71!, namely
Nn2m ~ Kb !
~88!
We have already evaluated L n (x) for x,0, see Eq. ~66!.
Now, we also need its value for x.0; we have
H n ~ kR ! J n ~ KR ! x ~ R;n 0 ! RdR,
`
u x 21u .a,
mPZ.
5M n 12 p in ein w W n ,
E
n0 ,
3L n2m ~ x 21! F n ~ x 2 ! dx 2 52e2im u ineia x 1 ,
3
c n ~ x 21 ,y 21!
3C ~ x 21 ,y 21! x ~ R 12 ;n 0 ! dx 2 dy 2
W n5
F m ~ x 1 ! 1n 0
x 1 2a
b,R 12,b 8
u x 21u ,a,
`
Here, we consider the effect of using a more complicated pair-correlation function, defined by Eq. ~15! in terms
of the function x (r;n 0 ). This function must decay rapidly to
zero as r→` and, in addition, x (r;n 0 )→0 as n 0 →0 for any
fixed r. For example, Bose and Mal ~1973! suggest using
x (r;n 0 )5e2r/L(n 0 ) , where the correlation length L(n 0 )→0
as n 0 →0. Other authors have supposed that x (r;n 0 )50 for
r.b 8 .b, where the radius b 8 may be taken as 2b; see, for
example, p. 1072 of Bose ~1996! or Eq. ~27! in Twersky
~1978!.
Proceeding as in Sec. IV B, we obtain Eq. ~57! with an
additional factor of @ 11 x (R 12 ;n 0 ) # in the integrand. Evaluating this equation for x 1 .,1b 8 , assuming that x (r;n 0 )
50 for r.b 8 , we obtain Eq. ~59! with M n2m replaced by
8 , where
M n2m
E
0,
was made. We shall show that use of Eq. ~88! leads to Twersky’s formula, Eq. ~4!.
Setting s51 and taking the conditional average of Eq.
~52! in the usual way, and looking for a solution in the form
of Eq. ~56! now leads to
D. Effect of pair-correlation function choice
M n8 5M n 1
H
J
1W n2m 50,
n in u in 2ia x
2i e
e
mPZ,
from which K is to be determined. This homogeneous system
for F n is Eq. ~33! in Bose and Mal ~1973! and it is a special
case of Eq. ~24! in Siqueira and Sarabandi ~1996!. Moreover,
the fact that W n 5o(1) as n 0 →0 means that the approximations for K obtained in Sec. IV C, namely Eqs. ~81! and ~86!,
are unchanged by the presence of x.
E. Reproducing Twersky’s formula
It is implicit in the work of Twersky ~1962! ~and others!
that the complications arising when a scatterer center is
closer to the boundary x50 than its radius are ignored. It
was pointed out by Lloyd and Berry ~1967! that, since all
scatterers are treated equally, ignoring the boundary-layer effects is equivalent to using a pair-correlation function with
the following property: if one scatterer is at (x 1 ,y 1 ), then no
other scatterer @with center (x 2 ,y 2 )] can occupy the infinite
strip x 1 2a,x 2 ,x 1 1a. Thus, instead of Eq. ~14!, the
choice
x.0
~92!
x,0.
Using these in Eq. ~91! gives
˜ eilx 1B̃eia x 52eia x ,
A
m
~87!
J. Acoust. Soc. Am., Vol. 117, No. 6, June 2005
H
2 ~ 2i! n e2in u ineia x
x.,1a,
mPZ,
where
`
˜ 5F 2 2in 0
A
F Z
m
m
a n52` n n
(
3
H
J
e2i~ l2 a ! a ei~ l1 a ! a i~ n2m ! u
T ,
2
e
l2 a
l1 a
`
H E
˜ 5 2n 0
Z ein u in
B
a n52` n
(
,
0
F n ~ t ! e2ia t dt1
iF n ei~ l2 a ! ,
l2 a
J
˜ 50 for all
and u T5 p 22 u in . Thus, l is to be found from A
m
mPZ.
As before, we write K 2 2k 2 5l 2 2 a 2 5 d 1 n 0 1 d 2 n 20
1¯. Hence,
e2i~ l2 a ! a ei~ l1 a ! a i~ n2m ! u
T
2
e
l2 a
l1 a
5
2ad2
1
2a
1
$ 122ia a2e2ia a ei~ n2m ! u T% 2 2
d 1n 0 2 a
d1
1O ~ n 0 ! .
C. Linton and P. Martin: Scattering by random arrangements of cylinders
3421
˜ 50 and neglecting terms that are O(n 2 ),
Substituting in A
m
0
we obtain
`
F m2
(
F nZ n
n52`
2
4in 0 d 2
d 21
J
H
4i
d1
1
in 0
a2
$ 122ia a2e2ia a ei~ n2m ! u T%
d 21
4a2
1
5
a2
4
a
~ 122ia a !
4
2
`
2ia a
e
`
( (
n52` s52`
Z n Z s ei~ s2n ! u T
$ e2ia a @ f ~ u T !# 2 2 ~ 122ia a !@ f ~ 0 !# 2 % .
Hence, if we let a a→0 in this formula, we recover Twersky’s ~erroneous! formula, Eq. ~4!.
V. CONCLUDING REMARKS
We have derived a two-dimensional version of the threedimensional Lloyd–Berry formula for the effective wave
number in a dilute random configuration of scatterers, using
methods that differ from those used by Lloyd and Berry
~1967!. Much remains to be done in order to validate the new
formula. Specifically, it should be possible to compare its
predictions with those obtained from full numerical simulations ~using Monte Carlo methods! and from experiments.
Some comparisons between Monte Carlo results and solutions of the infinite system ~87!, for various choices of x,
have been reported by Siqueira and Sarabandi ~1996!. They
used lossy cylinders, and found good agreement for low area
fractions, with little dependence on x.
Price et al. ~1988! have compared the predictions of Twersky’s formula, Eq. ~4!, with experimental results obtained
from sound propagation through three forests; they found
‘‘poor’’ agreement, but perhaps this could be attributed to
errors in the formula and the crude approximation of an actual forest by a random array of sound-hard circular cylinders. For a recent review of the quantification of attenuation
effects due to trees, see ~Attenborough, 2002!. Several other
experimental studies, in the context of fiber-reinforced materials, are cited in the paper by Verbis et al. ~2001!.
In three dimensions, there is an extensive literature on
comparisons between experiments, direct numerical simulations, and various theories, including the Lloyd–Berry formula; see, for example, Sec. 4.3.12 of Povey ~1997!, Hipp
et al. ~1999!, and references therein. For low volume fractions and properly modelled spheres, the agreement is generally good: according to Povey ~1997!, p. 133, there is ‘‘a
sizable body of evidence in support of the acoustic multiple
scattering theory.’’
3422
This work was supported by EPSRC Grant No. GR/
S35585/01.
50
for mPZ. Proceeding as in Sec. IV C, we obtain Eq. ~77!, as
before. Then, the O(n 0 ) terms give
d 25
ACKNOWLEDGMENT
J. Acoust. Soc. Am., Vol. 117, No. 6, June 2005
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